120 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

Table 6.1 Average values of daily simulated variables on [t,T]

Variable λ∓β 0 0.5 1 1.5

σV(%) 0.2 71.69 80.53 101.19 127.24

ρ 0.2 1.00 0.85 0.65 0.51

σV(%) 1 64.10 73.36 94.87 121.76

ρ 1 1.00 0.84 0.63 0.49

σV(%) 5 47.79 58.57 82.68 111.76

ρ 5 1.00 0.81 0.58 0.43

risk factor is a decreasing function ofλ. Second, the average correlation
coefficientρ(.) and global volatilityσV(.) are respectively decreasing and
increasing functions of the absolute value ofβ. Such a behavior is trivial
given thatρ(.) is the correlation between firm value and its idiosyncratic
risk factor. Third, average correlation coefficientρ(.) and global volatility
σV(.) are both decreasing functions ofλ.
We further setIt=0.1 and assumeω=0, which implies thatμX(t)= 0
whatevert. The diffusion of the idiosyncratic factor under the minimal
martingale measure then writes as:

dIt=

[

λ(ε−It)−

^2 I^2 t

σ^2 V(t,It)

(μV(t,It)−r)

]

dt+It

√

ItdWˆIt

withσ^2 V(t,Vt,It)=R(t,Vt,It), andμV(t,It)=λ(^0 I.t^5 −1)+^12 β(β−1)t−

1

(^2). Then,
the related average stochastic variance σ ̄^2 V conditional on Gt reads
σ ̄^2 V=^1 τ
∫T
t σ
2
V(s,Is)ds=
2 β^2
τ
(√
T−
√
t
)
+
2
τ
∫T
t Isdswhenρ(s,Is) is zero.
From formula (6.10), debt computation requires the estimation of the
call’s price, which we realize with Monte Carlo simulation and antithetic
variables-based accelerators for variance reduction principle (Jäckel, 2002;
Ripley, 1987; Rubinstein, 1981). Letnsimbe the number of simulations and
CBS,k(.) be the call’s price of thek-th simulation. Then, the estimated equity
value conditional on current information is the arithmetic mean of simulated
variables, namely:
E(Vt,τ)=EPˆ
[
CBS
(
τ,r,Vt,B,
√
σ ̄V^2
)∣
∣
∣
∣Ft
]

1
nsim
nsim∑
k= 1
CBS,k
(
τ,r,Vt,B,
√
σ ̄^2 V
)
+CBS,k
(
τ,r,Vt,B,−
√
σ ̄^2 V
)
2
We realize monthly simulations withr=8%,B=13,Vt=52 whereVand
Bare expressed in billions of dollars. Our examination then uses varying